Acoustic waves: should they be propagated forward in time, or forward in space?

TL;DR

This paper compares spatial and temporal propagation methods for acoustic waves, analyzing their advantages, limitations, and suitability for different wave equations and physical scenarios.

Contribution

It provides a detailed comparison of space and time propagation techniques, introducing a unified framework based on directional factorization and slow evolution approximation.

Findings

Spatial propagation handles dispersion and interfaces efficiently.

Temporal propagation accurately models reflections and scattering.

Both methods can be unified through directional factorization.

Abstract

The evolution of acoustic waves can be evaluated in two ways: either as a temporal, or a spatial propagation. Propagating in space provides the considerable advantage of being able to handle dispersion and propagation across interfaces with remarkable efficiency; but propagating in time is more physical and gives correctly behaved reflections and scattering without effort. Which should be chosen in a given situation, and what compromises might have to be made? Here the natural behaviors of each choice of propagation are compared and contrasted for an ordinary second order wave equation, the time-dependent diffusion wave equation, an elastic rod wave equation, and the Stokes'/ van Wijngaarden's equations, each case illuminating a characteristic feature of the technique. Either choice of propagation axis enables a partitioning the wave equation that gives rise to a directional factorization based on a natural "reference" dispersion relation. The resulting exact coupled bidirectional equations then reduce to a single unidirectional first-order wave equation using a simple "slow evolution" assumption that minimizes effect of subsequent approximations, while allowing a direct term-to-term comparison between exact and approximate theories.